Let's analyze the transformation applied to the function [tex]\( g(x) = x^2 \)[/tex] to obtain the function [tex]\( h(x) \)[/tex].
The transformation given is:
[tex]\[ h(x) = g(x - 3) \][/tex]
This means wherever there was an [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex], it is replaced by [tex]\( x - 3 \)[/tex].
To understand the transformation, recall how function transformations work:
1. Horizontal Shifts:
- A function [tex]\( g(x - a) \)[/tex] represents a horizontal shift to the right by [tex]\( a \)[/tex] units.
- A function [tex]\( g(x + a) \)[/tex] represents a horizontal shift to the left by [tex]\( a \)[/tex] units.
2. Vertical Shifts:
- A function [tex]\( g(x) + a \)[/tex] represents a vertical shift up by [tex]\( a \)[/tex] units.
- A function [tex]\( g(x) - a \)[/tex] represents a vertical shift down by [tex]\( a \)[/tex] units.
Given our function transformation:
[tex]\[ h(x) = g(x - 3) \][/tex]
This fits the horizontal shift transformation. Specifically, [tex]\( x - 3 \)[/tex] indicates a shift to the right by 3 units.
So, the correct statement describing the difference between the graph of [tex]\( h \)[/tex] and the graph of [tex]\( g \)[/tex] is:
A. The graph of [tex]\( h \)[/tex] is the graph of [tex]\( g \)[/tex] horizontally shifted right 3 units.
